On the Integrability of Strongly Regular Graphs

Abstract

Koolen et al. showed that if a connected graph with smallest eigenvalue at least −3 has large minimal valency, then it is 2-integrable. In this paper, we will prove that a lower bound for the minimal valency is 166.

Problems on Graphs with Fixed Smallest Eigenvalue

“…(ii) It is known that κ 1 is at least 166 by results of Koolen and Munemasa [59] and Koolen, Rehman and Yang [61].…”

“…At the end of this survey, we give some problems for discussion. In [61], Koolen et al showed that the complement of the McLaughlin graph has smallest eigenvalue −3 and can not be 2-integrated, but it is 4-integrable. It is not clear whether any graph with smallest eigenvalue at least −3 is always 4-integrable.…”

On graphs with smallest eigenvalue at least −3 and their lattices

“…Note that M cL is the unique strongly regular graph with parameters (275, 162, 105, 81), as the McLaughlin graph is uniquely determined by its parameters (See [6]). In [11], Koolen et al showed that M cL is not 2-integrable. Let H be the 3-point cone over M cL, that is, H is obtained by adding 3 new vertices to M cL and joining these three vertices to all the vertices in M cL and to each other.…”